On the Normality of Arithmetical Constants
Jeffrey C. Lagarias
CONTENTS
Bailey and Crandall recently formulated "Hypothesis A", a genprinciple to explain the (conjectured) normality of the bi-
1. Introduction
era|
2. Radix Expansions
nary expansion of constants like n and other related numbers,
3. Perturbed Radix Expansions
or more generally the base b expansion of such constants for an
4. BBP-Numbers and Hypothesis A
integer b > 2. This paper shows that a basic mechanism under-
5. Special Values of G-Functions
.
. .u
j r * u
/ / - • *
r
6. Invariant Measures and Furstenberg's Conjecture
7. Concluding Remarks
Acknowledgements
'Y'ng their principle, which is a relation between single orbits of
two discrete dynamical systems, holds for a very general class
.
.
.
.
. .
, .
of representations of numbers. This general class includes numb e r s for w h j c h t h e c o n d u s i o n o f Hypothesis A is not true. The
paper also relates the particular class of arithmetical constants
References
treated by Bailey and Crandall to special values of G-functions,
and points out an analogy of Hypothesis A with Furstenberg's
conjecture on invariant measures.
1. INTRODUCTION
Much is known about the irrationality and transcendence of classical arithmetical constants such as TT,
e, and £(n) for n > 2. There are general methods which in many cases establish irrationality or
transcendence of such numbers. In contrast, almost nothing is known concerning the question of
whether arithmetical constants are normal numbers
to a fixed base, say 6 = 2. It is unknown whether
any algebraic number is normal to any integer base
b > 2. Even very weak assertions in the direction
of normality are unresolved. For example, it is not
known whether arbitrarily long blocks of zeros appear in the binary expansion of y/2.
Bailey and Crandall [2001] formulated what they
called "Hypothesis A", which provides a hypothetical general principle to explain the (conjectured)
normality to base 2 of certain arithmetical constants
such as 7T and log 2.
Hypothesis A. Consider a positive integer b > 2 and
8
1
P r i m a r y U K 1 6 ; SeC ndary
^ e ^ D O ^ ! ^ ^ ^
'
°
'
Keywords: dylamical systems, invariant measures, G-functions,
poiyiogarithms, radix expansions
a raUonal
function RW = P(x)/q(x) E Q(x) sueh
that
degp(z) < deg q(x) and that g(x) has no nonnegative integer roots. Let 6 = X^oP( n )IQ( n )b n ,
© A K Peters, Ltd.
1058-6458/2001 $0.50 per page
Experimental Mathematics 10:3, page 355
356
Experimental Mathematics, Vol. 10 (2001), No. 3
and define a sequence {yn : n > 0} by setting y0 = 0,
| ^n'
(mod 1)
(1-1)
Q\n)
m^
,1,7
7
n -, 7
,. .,
lhen this sequence either has finitely many limit
. .
•
•/. 7 j • , 7 . i
71
points or is uniformly distributed mod 1.
y
—by
This hypothesis concerns the behavior of a particular orbit of the discrete dynamical system (1-1).
Assuming Hypothesis A, Bailey and Crandall deduced that the number 9 either is rational or else is
a normal number to base b\ these correspond to the
two possible behaviors of the sequence {yn : n > 1}
allowed by Hypothesis A, see Theorem 4.2 below.
Proving Hypothesis A appears intractable, but it
seems useful in collecting a number of disparate phe-
We now summarize the contents of the paper in
more detail. In Sections 2 and 3 we give the dynam*ca* c o n n e c t i o n underlying Hypothesis A. In Section 2 we review radix expansions to an integer base
6 > 2 treated as a discrete dynamical system acting
J
J
~
,
°
r
on the interval 0,1 . The radix expansion of a real
*
->•><,
r
number 9 is described by an orbit or a dynamical
system, the ^-transformation Th(x) = bx (mod 1),
studied by Renyi [1957] and Parry [I960]. For a
given number 9 its 6-expansion can be computed
from the iterates of this system
Xn+1
with
sion
= bXn
(mod ^
initia j[ condition x0 = 9 (mod 1). The b-expano f a r e a i n u m b e r 9 G [0,1] is
oo
nomona together under a single principle. A formulation in terms of dynamical systems is natural, because the property of normality is itself expressable
in terms of dynamics of an orbit of another dynamical system, the 6-transformation, see Section 2. The
basic mechanism rendering Hypothesis A useful is a
relation between particular orbits of these two different dynamical systems.
This paper provides some complements to the resuits of Bailey and Crandall. It shows that the relation between particular orbits of two discrete dynamical systems underlying Hypothesis A is valid
very generally, in that it applies to expansions of
real numbers of the form, 9 = Y^=ienb~n,
with
en arbitrary real numbers with en ->• 0 as n ->• oo;
see Theorem 3.1. Every real number has such an
expansion. Hypothesis A is not true in such generality, so in order to be valid Hypothesis A must
be restricted to apply only to expansions of some
special form. Bailey and Crandall do this, formulating Hypothesis A only for a countable class of
arithmetical constants which in the sequel we call
BBP-numbers. It does not seem clear what should
be the "optimal" class of arithmetical constants for
which Hypothesis A might be valid. The remainder
of the paper discusses various mathematical topics
relevant to this issue. We relate BBP-numbers to
the theory of G-functions and characterize the subclass of BBP-numbers which are "special values" of
G-functions. We also compare Hypothesis A to a
conjecture of Furstenberg in ergodic theory, and this
suggests some further questions to pursue.
a _ v~^ , , -j
^ J
in w h i c h t h e
. H h digit is defined by dj := L^j-iJ •
Section 3 we suppose the given real number 9 is
expressed as
J^
2^Sn
'
0-^)
In
n=1
in which en is any sequence of real numbers with
sn —> 0 as n —> oo. To this one can associate a
perturbed b-expansion associated to the perturbed btransformation
2M+1 =
hyn + £
(1 3)
^+ 1 ( m o d *)'
~
condition
y
G
[0,1).
The
restarting w i t h a n initial
0
c u r r e n C e (1-3) is an infinite sequence of maps which
change at each iteration. Associated to this recurb-expansion
r e n c e i s t h e perturbed
4_ /9 — V^ J h~i
3
^
in w h i c h t h e
J - t h digit is defined by
T
i,
, i
3
3
J
Choosing the initial condition y0 — 0 gives the perturbed 6-expansion of 9. The mechanism underlying
the approach of Bailey and Crandall is that the the
6-expansion of 9 and the perturbed 6-expansion of 9
are strongly correlated in the following sense: The
orbit {yn : n > 0} of the perturbed 6-transformation
Lagarias: On the Normality of Arithmetical Constants 357
with initial condition y0 — 0 asymptotically approaches the orbit {xn : n > 0} associated to the
6-transformation with initial condition x0 = 0 — [6\
(Theorem 3.1). In particular, the orbits {xn : n > 0}
and {yn : n > 0} have the same set of limit points,
and one is uniformly distributed (mod 1) if and
only if the other is. This implies that the perturbed ^-expansion of #, though different from the
6-expansion of #, has similar statistics, in various
senses. This connection is quite general, since every
real number 6 has representations of the form (1-2).
In Section 4 we consider the particular class of
arithmetical constants treated in [Bailey and Crandall 2001], consisting of the countable set of 9 given
by an expansion (1-2) with b > 2 an integer and
en — p{n)/q{n), where p(x), q(x) G Z(x) and q(n) ^
0 for all n > 0. We call such numbers BBP-numbers,
and call the associated formula
00
/ \
6=
b n
Yl ~7n) ~ '
n=l
satisfy all but one of the properties required to be a
special value of a G-function defined over the base
field Q. We then show that a BBP-expansion to
base b corresponds to a special value of a G-series
at z — 1/b if and only if the denominator polynomial q(x) (in lowest terms) factors into linear factors over the rationals (Theorem 5.4). We show
that if all the roots of q(x) are distinct, then such
special values are either rational or transcendental, using Baker's results on linear forms in logarithms, in Theorem 5.5, a result also obtained by
Adikhari, Saradha, Shorey and Tijdeman [Adhikari
et al. > 2001]. We summarize other known results
about irrationality or transcendence of special values of G-functions of the type in Theorem 5.4. It is
interesting to observe that every one of the examples
given in [Bailey and Crandall 2001] is a special value
of a G-function. Many other interesting examples of
such constants were known earlier; for example, D.
H Lehmer 1975 p 1 3 9 observed that
*
t
'* ]
^'
a BBP-expansion to base b of 6. These numbers
V^
JL
are named after Bailey, Borwein and Plouffe [Bai^ (n+l)(2n+l)(4n+l)
3'
ley et al. 1997], who demonstrated the usefulness of
_
,
.
x
TX
A . _
,
, ,.
, ,
,
, y. ,
, w •
In Section 6 we compare Hypothesis A with a consuch representations (when degp(x) < aegq(x)) in
.
,.
n^
,
,.
i
r
i
1
lecture
01
Furstenbere
m
ergodic
theory, concerning
7
computing base 6 radix expansions or such numbers.
_
, . - , . .
r
^T
., -nn-n u 1. • ^
JJ-X- 1 measures that are ergodic for the joint action of
We consider BPP-numbers having the additional
. . .. . _ . .
.
.
.
^ . ^.
,
,
,
/ \ r ^i •
T^- •
two multiplicatively independent o-transrormations.
restriction degp(x)x < degg(x), for this condition is
^ .
:
.
f ..
. .
.
.
,
' . ^n
r,
Botn conjectures have similar conclusions, though
necessary and sufficient for en —>> 0 as n —> 00, so
.
.
..
. . .
. .
^ i ^
i, r o ,- o
i mi
1
there seems to be no direct relation between their
that the results of Section 3 apply. The number. . . .
,,
,
.-.-.^
.
. , ! , , !
hypotheses. Bailey and Crandall have found examL.
1
theoretic character of BPP- numbers is that they
i
P .,
. ,
n-ni i
. , ,
.
,
. t x rr
,.
pies of arithmetical constants which have the propt i
are special values (at rational points) of functions
_. . ^ ^ ^ ,
, . ,. . ,
er
,. r .
1
T
i-rrx-i
xt y °f being BBP-numbers to two multiphcatively
satisfying a homogeneous linear differential equation
. .
.
.
_. .
.
. _.
.^, . ^
,
.,
rn - x TTT i • n
independent bases. 1ms suggests that one should
with integer polynomial coefficients. We derive the
. .r r .
...
°
1. i 1
,, r r . „
.~
1 n ,1 , TT
,i • A •
look tor further conditions under which the two conresult of Bailey and Crandall that Hypothesis A 1m.
1 . 1 1 1
,.
,, ,
, ~ .,_,
!
T lectures are more directly related,
plies that such 6 either are rational or are normal
.
_
,
, , .
, TT7
T
n
,
, /rni
An \ ™ •
i^ ^
In oection 7 we make concluding remarks. We
x ,
numbers to base b (lheorem 4.2.) 1 nis result makes
.
..
.. .
_ .
.
_
. r- 2- i ,. n 1 -i • i. i /
•
i T-.T-»I-I
describe an empirical taxonomy of various classes of
itA of interest to find criteria to determine when BBP..
. .
.r
.
.
,
, ,.,
.,
,
arithmetical constants, and formulate some alternanumbers are irrational, which we consider next.
.
..
._
,. r
, , T^T^T^
i
, ,i ,i
tive classes of arithmetical constants as candidates
T n
In bection 5 we relate BBP-numbers to the tne. . . . __
. .
r
r ^ r >ii
,-,1
ii
ory ot G-tunctions, and characterize tne subclass
of BBP-numbers which are "special values" of Gfunctions. The subject of G-functions has been extensively developed in recent years (see [Andre 1989;
Bombieri 1981; Dwork et al. 1994]) and the special values of such functions can often be proved
to be irrational [Bombieri 1981; Chudnovsky 1984;
Galochkin 1974]. We observe that BBP-numbers
tor inclusion m Hypothesis A.
2. RADIX EXPANSIONS
W e nQw consider radix expansions to an integer base
> 2 Such expansions are obtained by iterating the
.
b transformation
&
Th{x) = bx (mod 1).
358
Experimental Mathematics, Vol. 10 (2001), No. 3
Given a real number x0 G [0,1), as initial condition,
we produce the sequence of remainders
,
, iN
/
xn+1 = bxn (mod
1),
_n
with 0 < xn+1 < 1. That is,
~~
xn+1 = bxn - dn+1
(2-1)
where
Recall that the uniform measure or Lebesgue measure /iLeb on [0,1] is the unique absolutely continuous invariant measure for the ^-transformation Tb.
Definition 2.2. A real number 9 G [0,1) is normal
, ,
, ., . ,
-,. .,
to base b if tor every m > 1 every digit sequence
d±d2 • • • dm G {0,
1, . . . , d-l}m occurs with limiting
m
frequency 6~~ , as given by the invariant measure
MLeb-
dn+i = d n + i 0 o ) = \bxn\ G { 0 , 1 , . . . , 6 - 1}
is called the n-th digit of 9. The forward orbit of
x0 is O+(* 0 ) = {xn : n ^ 0} and we call {xn} the
remainder sequence ot the o-expansion. Iterating
^ .
. ,,
/rk ^x
(2-1)
v
y n + 1 times yields
x n + 1 - 6 n + 1 ^ 0 - d n+1 - bdn
6 n di.
(2-2)
Dividing by bn+1 yields
Recall that
//Leb({^o : di(a 0 ) • • • dn(x0) = did 2 • • • d m }) = b~m.
.g ^
^
ft
known ^
for
ft
thfi ^
of
.-,
,,
,
,
.
,
,
,
,,
,
n
rrk
£
T
a G L0,1J that are normal to base b nas mil Lebesgue
measure.
The properties of the digit expansion {dn{&): n > 1}
can be extracted from the remainder sequence {xn}.
The
following
r e s u l t is w e l l k n o w n .
n
x
=
\ ^ db~j — b~n~1x
Theorem 2.3. Consider an integer base b > 2 and a
real number 6 G [0,1].
j=1
Letting n -» oc yields the b-expansion of ^ 0 ,
oo
x0 = J2dj(xo)b-j,
j=i
(1) <^ is digit-dense
to base b if and only if its remainder sequence {xn(6) : n > 1} to base b is
dense in [0,1].
(2) 9 is normal to base b if and only if its remainder
which is valid for 0 < x0 < 1. For 9 G R we take
xo = 6- [6\ and do(9) = [6\ € Z, thus obtaining
,,
, ,.
n J (t\\ . Y^ J /ML-?
v = ao\v) + > dj(v)b ,
r^
which is called the b-expansion of 9. Note that (2-2)
Slves
sequence
uni rml
^ ^ to base b is
f° V
M
6ttted m [0 1]
, * f"
> . ,. ,
.
.,
,
(3) u has an eventually periodic b-expansion if and
only if its remainder sequence {xn : n > 1} to
base b has finitely many limit points. This condi.
f
tzon /zo/as ^/ ana on/y z/ | x n : n > 1} eventually
enters a periodic orbit of the b-transformation,
for some m ? p > L T / i e 5 e e g ^ v .
Le^ Xm = x^^
^
: n
a/en£ conditions hold if and only 9 is rational.
n
n
xn = b x0 (mod 1) = b 9 (mod 1)
(2-3)
in this case.
The following property of 9 concerns the topological dynamics of the 6-transformation for its iterates.
Definition 2.1. A real number 9 G [0,1) is digit-dense
to base b if, for every m > 1, every legal digit sequence of digits of length m occurs at least once as
consecutive digits in the 6-expansion
oo
9 = y ^ dn(9)/3~n.
n=i
The following property of 9 concerns the metric
dynamics of the ^-transformation for its iterates.
Proof. (1) The set
J(did2 • • -rfm) := {^G [0,1] :d1{9) • • -d m (e) = d r • -d m }
ig a h a l f . o p e n i n t e r v a l [flj a + 6 - m ) o f l e n g t h 6 -m ? a n d
t h e bm i n t e r v a i s p a r t i t i o n [0,1]. Digit-denseness imP 1 1 ^ there exists some xk G I{dx • • • dm). This holds
for
all m > 1 and generates a dense set of points.
(2) If {xn : n > 1} is uniformly distributed (mod 1),
then the correct frequency of points occurs in each
interval I(di • • • d m ), and this proves normality of 9.
For the converse, one uses the fact that I{d\ • • • dm)
is a basis for the Borel sets in [0,1).
(3) The key point to check is that if the limit
set of {xn : n > 1} is finite, then this finite set
Lagarias: On the Normality of Arithmetical Constants
359
forms a single periodic orbit of the 6-transformation,
where, for n > 0,
and some xn lies in this orbit. We omit details; see
~ _ .
.
n+1
[Bailey and Crandall 2001, Theorem 2.8].
•
~ L Vn + £ n + l J G
. , , , . , ,
i
i. , ,i
is the (n+l)-st digit of the expansion. The digit
nj
Remark. Most ot the results above generalize to the
~
,
_
7/ .
r
/^-transformation 7>(*) = /fe (mod 1) for a fixed Sequm'e>
= ^ o ) a *d ™ ^ ; se^ence {yn :
, ^
- ,,
x v J T- TI
r-ir.^^1
n > If depend on t h e initial condition y0.
real p > 1; these maps were studied by Parry 1960 .
. , , , xl .
. ,,
,.
n
•
A
r
Associated to this map is the notion or a p-expansion
for any real number 0, in which the allowed digits
are { 0 , 1 , 2 , . . . , \J5\}. Not all digit sequences are al-
~~ _J „ „ _ . _ _
_ *
£n —>
) 0, lor all sufficiently large n, one has
,
Mow (6-2) iterated n + 1 times yields
j .
n
-
i , n
x r n
J J- -x
lowed in p-expansions, but the set of allowed digit
sequences was characterized by Parry; see [Flatto
et al. 1994] for other references. One defines a number 6 to be digit-dense to base (3 if every allowable finite digit sequence occurs in its /3-expansion. There
5
\
, ,
1
,.
.
3. PERTURBED RADIX EXPANSIONS
Let b > 2 be an integer, and let {sn : n > 1} be an
arbitrary sequence of real numbers satisfying
n
Set
00
0 = ^(6, {Sn}) '-= 2_^£n^
-
(3-1)
n=1
We can study the real number 6 using a perturbed
b-expansion associated to the sequence {en}.
The perturbed b-transformation on [0,1) is the recurrence
(modi),
fc+1=%o
+ £,+1
with 0 < y n + i < 1 and with given initial condition
?/o. That is,
y n + i = byn + 6 n+ i - J n +i,
dn £ {—1, 0, 1, . . . , 6—1, 6}.
.
v
J
'
2/n+i = en+1 + bsn-\
J2^ „
h6ngi + 6 n + 1 7/ o -^Jd n + 1 _ j 6 7 '.
j=0
.1
Dividing by 6 n+1 yields
. ,
is a unique absolutely continuous invariant measure
d// of total mass one for the /3-transformation, and
one defines a number ^ to be normal to base (3 if
every finite block of digits occurs in its /^-expansion
with the limiting frequency prescribed by this invariant measure. With these conventions, Theorem 2.3
remains valid for a general base /?, except that Theorem 2.3(3) must be taken only as characterizing
eventually periodic orbits of the /^-transformation.
That is, the final assertion in (3) that 9 is rational
must be dropped; it does not hold for general (3.
For results relating normality of numbers in different real bases 0, see [Brown et al. 1997].
n->oo
bmce
(3-2)
°
JltJ; ^
z^ j
j=1
.
=
^i
2-^£i
+ \Vo ~ b
J/n+i)«
j=1
Letting n —> oo yields the perturbed b-expansion
oo
y0 + 9 = V^ dj(yo)b~j,
j=i
<
y
<
1.
We
write yn = yn(y0) for the
v a l i d for 0
0
r e m a i n d e r s e q uence in (3-2)
{^(0) : n > 1} for ^
T h e perturbed
b.expansion
_
g i y e n b y ( 3 1 } i g o b t a i n e d b y c h o o S ing the initial
condition y0 = 0, i.e., d*n(9) := dn(0). We also have
the perturbed remainders {y^(^) : n > 1} given by
y*n(9) = j/ n (0).
The main
N e r v a t i o n of this section is that the
remainders of the perturbed 6-expansion of such 9
are related to the remainders of their 6-expansion.
Theorem 3.1. Let b > 2 be an integer and let 6 :=
S^Li £nb~n\ where en are real numbers with en —> 0
as n -> ex). Let {y*(^) : n > 1} denote the associatedperturbed remainder sequence of #, and {xn{6) :
n > 1} ^/ie remainder sequence of its b-expansion.
If
~
n :=
2^ £n+i
'
j=1
i . m - A W
The
orbits
+t .
(modi).
(3-4,
ixn(°) : n > !} a n r f (2/n(#) • ^ > 1}
asymptotically approach each other on the torus T =
M/Z a5 n -> oo.
360
Experimental Mathematics, Vol. 10 (2001), No. 3
Proof. Since y0 = 0, formula (3-3) gives
n+1
yn+1 = ^
bn+1~jSj
(mod 1).
j=1
Now
J^
.
™
b-6 = Y, bn^ej = £ ir-'e, + tn.
3= 1
We next consider perturbed 6-expansions having
a finite number of limit points, and show that they
correspond to rational 0.
Theorem 3.3. Let b > 2 be an integer and let 6 =
]C^Li £nb~n with en a sequence of real numbers with
en -> 0 as n —> oo. The following conditions are
equivalent.
3= 1
Thus
— Vn -r n vmo j .
)
For the 6-expansion, (2-3) gives bn9 = xn (mod 1),
and combining this with (3-5) yields (3-4).
Since en -> 0 as n -> oo, we have tn -> 0 as
n -^ oo. Thus |xn((9) — 2/*(<9)| —> 0 on T as
n -» oo. Note that on T = R / Z the points £ and
Lemma 3.2. Let {xn : n > 1} and {yn : n > 1} be any
too sequences in [0,1] wi£/i a;n = y n + Sn (mod 1)
y
^z^/i on -> 0 as n -> oo.
_7
(1) T/ie sequences \xn : n > 1} ana {yn : n > 1}
Ziave the same sets of limit points, provided the
endpoints 0 and 1 are identified.
(2) The sequence {xn : n > 1} is uniformly distributed (mod 1) if and only if {yn : n > 1} is
uniformly distributed (mod 1).
Proof. (1) This is clear since xn. -> ^ implies i/n. ^
^ and vice-versa, except at the endpoints ^ = 0
or 1, which, by convention, we identify as the same
•nt
(2) This is well known; see [Kuipers and Niederreiter 1974, Theorem 1.2, p. 3].
•
One can compare the fo-expansion {dn(6) : n > 1}
and the perturbed 6-expansion {d*n(6) : n > 1} of
such 0 We have
dn[0) = Lten-iJ,
< ( 0 ) = [byn-i+£n\ = L&(x n _i-t n _i(mod 1)) +en\.
Since tn .-> 0 and en -+ 0 as n -> oo, one expects
that "most" digit values of the two expansions will
agree, in the sense that dn{6) = d* (0) for "most"
sufficiently large values of n. (This is an unproved
heuristic statement. It is an open problem to prove
that a natural density-one proportion of all n have
dn(8) — rf* (0).) However there is still room for there
to be infinitely many n where dn(0) ^ d* (0).
0) 0 € Q.
(ii) The remainders {y* (0) : n > 1} o/ t/ie perturbed
b-expansion of 0 have finitely many limit points
in [0,1].
(iii) The orbit {y*(0) : n > 1} of the perturbed btransformation asymptotically approaches a periodic orferf
{xfc : 0 < k < p}
of the b-transformation,
Tb(xp) = x0 and for 0 <
*^ N
, r / J i\
l/nW = ^ + *n (mod 1)
with Tb(xk) = x^+i and
j < p - 1. 77ia£ is,
•*
• / ^ \ /-> n
if n = j (mod p) (3-6)
J
mt/i 5 n —>> 0 as n —>• oo.
Proof, (i) => (ii). By Theorem 2.3 if 0 E Q the remainders {xn(6) : n > 1} of the ^-transformation
have finitely many limit points. By Theorem 3.1
and Lemma 3.2 we conclude that {y*(0) : n > 1}
has the same set of limit points.
(U) = ^ (i11)' B ^ T h e o r e m 3.1 and Lemma 3.2 the
limit
P o i n t s o f ivM)
• n > 1} are the same as
K W ^ > ! } • % Theorem 2.3 such limit points
must form a periodic orbit of the 6-transformation.
( m ) = * ^' T h e v a l u e s {»«(«) • ^ > 1} h a v e l i m i t
oints the
P
P e r i o d i c o r b i t ixj = 1 < J < n} of Th. By
Theorem 2.3, it follows that 0 G Q.
D
^marks. 1. Any real number 0 has some perturbed
6-expansion that satisfies the hypotheses of Theorem 3.1, so in a sense these expansions are completely general. It follows from Theorem 3.3 that
Hypothesis A cannot be valid for all such 0, since
there exist
irrational 0 that are not normal numbers.
2. The rationality criterion of Theorem 3.3 is not
directly testable computationally, unless all en = 0
for n > n 0 ; the latter case is essentially the same as
that of a ^-transformation. When infinitely many
en are nonzero, then the points {y* (0) : n > 1} stay
outside the periodic orbit for infinitely many values
Lagarias: On the Normality of Arithmetical Constants
361
of n, and the role of the {£n} is to compensate for
the expanding nature of the map T(x) — bx (mod 1)
by providing negative feedback to push the iterates
closer and closer to the periodic orbit.
We now formulate two hypotheses, whose conclusions are in terms of topological dynamics and metric dynamics, respectively. The second of these is
Hypothesis A of [Bailey and Crandall 2001].
(3) Theorem 3.1 does not extend to ^-expansions
for noninteger /?. One can consider
W eak
00
0 = 9((3, {en}) := ^ e n / T n .
71=1
and define an associated perturbed /3-transformation
. . .
. ' .
.
TT
ln t h e obvious way. However when b is n o t a n m t e .
.
_ _.
i P 'i
i i i rt
ger the ana ogue of Theorem
3 1 faxls to hold, since
(3-5) 1S no longer valid. In particular, Theorem 3.1
does not extend to rational/? = 6/a > 1, with a > 1.
;
4. BBP-NUMBERS AND HYPOTHESIS A
Dichotomy Hypothesis. Let there be given a perb-transformation with en = p{n)/q{n), where
turbed
x
p( )-> <l(x) € 2 [a;] anddegq(x) > degp(x). Then the
orbit {yn : n > 1} for 9{b, {sn}) either has finitely
many limit points or else is dense in [0,1].
^
^. • *
.. ^ . T ± ±i
i
Strong Dichotomy Hypothesis. Let there be given a per, ,
,
.
,.
.,,
/ \/ / \ i
turbed7 7b-transformation with en — pin) a in), where
^
^
£
>
l
^ ^
.
^
>
^
^
{
1}
{ {
}
« . 7
.
.
./
; »...».,
7. .
finitely many limit points or is uniformly distributed
on [0,1]. Equivalently, in measure theoretic terms,
the measures
N
We consider expansions of the following special form.
Definition 4.1. A BBP-number to base b is a real number 6 with a representation
oo / x
6 = V^ ^-~^-b~n,
(4-1)
n
n = 1 Q\ )
in which b > 2 is an integer and p(x), q(x) G Z[x]
are relatively prime polynomials, with q(n) ^ 0 for
each n G Z> 0 . We call (4-1) a BBP-expansion to
base b.
The name BBP-number refers to Bailey, Borwein
and Plouffe, who introduced this class of numbers
[Bailey et al. 1997, p. 904], proving that the d-th
digit of such a number is computable in time at most
O{d\og }d) using space at most O(log }d). (Here
"computing the d-th digit" is understood to mean
computing an approximation to bd9 (mod 1) that
is guaranteed to be within a specified distance to
bd9 (mod 1); usually this determines the d-th digit,
but it may not, near the endpoints of the digit interval.) In other words, computing digits of a BBPnumber is a problem of complexity class 5(7*, a subclass of SC [Johnson 1990, p. 127].
We mainly consider BBP-numbers that satisfy the
extra condition
degq(x) > degp(x).
(4-2)
This condition guarantees that en =p(n)/q(n) -> 0
as n —> oo, which makes Theorem 3.1 applicable.
_ J_ v ^ c
k==1
converge in the vague topology as N -> oc to a limit
measure JJL, which is an invariant measure for the btransformation, and which is either a measure supported on a finite set or else is Lebesgue measure on
LU' J*
The following conditional theorem is a central result
o f [Bailey and Crandall 2001]:
Theorem 4.2. Let 9 be a BBP-number to base b whose
associated BBP expansion satisfies
( 1 ) If the
eUher
Weak
mUond
( 2 ) If the Strong
either
Proof
-
rational
degg(x) > degp(x).
Hypothesis is true, 9 is
Dichotomy
digit-dense
to base b.
or
Hypothesis is true, 9 is
Dichotomy
or a normai
number
to base bm
The
condition degq(x) > degp(x) guarantees
^n = p(n)/q(n) -> 0 as n -> oo. Thus Theorem
3- 1 applies to the BBP-number
oo , ,
6 =
Y1 (n)b~n'
that
n=1
(1) By the Weak Dichotomy Hypothesis, the limit
Set
° f ^ ^ : n > 1} is dense in [0,1]. Therefore
Lemma 3.2(1) implies that the 6-expansion remainders {xn(9) : n > 1} are dense in [0,1]. Theorem
2.3(1) then shows that 9 is digit-dense.
362
Experimental Mathematics, Vol. 10 (2001), No. 3
(2) By the Strong Dichotomy Hypothesis, the sequence {y*(0) : n > 1} is uniformly distributed in
[0,1]. Therefore Lemma 3.2(2) implies that {xn(0) :
n > 1} is uniformly distributed in [0,1]. Now 6 is
normal to base b by Theorem 2.3(2).
•
Many examples of BBP-numbers satisfying (4-2)
where the associated real number 6 is known to be
irrational are given in [Bailey et al. 1997; Bailey and
Crandall 2001]. For example for various b one can
obtain 7T, log 2, C(3) etc. Bailey and Crandall also
observe that C(5) is a BBP-number, to base b = 2 60 ,
but it remains an open problem to decide if C(5)
is irrational. All the examples they give of BBPnumbers are actually of a special form: they are
"special values" of G-functions defined over Q, as
we discuss next.
5. SPECIAL VALUES OF G-FUNCTIONS
Definition 5.1. A power series
f(\
—V ^
~^
n
There is an extensive theory of G-functions; see
[Bombieri 1981; Andre 1989; Dworket al. 1994]. For
the general definition of a G-function over an algebraic number field K see [Andre 1989, p. 14; Dwork
et al. 1994]. G-functions have an important role
in arithmetic algebraic geometry, where it is conj e c t u r e d t h a t t h e y a r e e x a c t l y t h e s e t o f s o l u tions
o y e r Q^ o f a g e o m e t r i c differential equation over
Q ? a g d e f i n e d i n [ A n d r 6 1 9 g 9 j p 2] I n a n y c a g e
i t ig k n o w n t h a t t h e ( m i n i m a l ) homogeneous linear
d i f f e r e n t i a l e q u a t i o n satisfied by a G-series is of a
yery restricted kind:
it m u s t h a v e r e g u l a r
points? a n d these m u s t
b y a regult of R a t z
gee
bieri a n d g p e r b e r 1982]
singular
a l l h a v e r a t i o n a l exponents,
[ B o m b i e r i 1 9 g l i p- 46; B o m .
( T h e g r o w t h c o n d i t i o n (iv)
p l a y s a c m d a l r o l e i n o b t a i n i n g t h i s r e s u l t . ) I t fol_
£ _ s e r i e s analytically continues to a multivalued function on P 1 (C) minus a finite number of
singular points [Dwork et al. 1994, p. xiv]. We call
this multivalued function a G-function.
I t i s known that the set 9K of G-series defined
over a number field K forms a ring over K. under
addition and multiplication, which is also closed under the Hadamard product
lows that a
oo
defines a G-series over the base field Q if the following conditions hold.
(i) Rational coefficients condition. All the an are rational, so we may write an — Pn/Qn, with p n , qn G
Z such that (p n ,g n ) = 1 and qn > 1.
(ii) Local analyticity condition. The power series f(z)
has positive radius of convergence r<xn and for
each prime p the p-adic function
^
f rz\ ._ V ^ a
^Q
n
z
viewed with an G Q C Q p , has positive radius
of convergence rp in C p , the completion of the
algebraic closure of Qp.
(iii) Linear differential equation condition. The power
series f(z) formally satisfies a homogeneous linear differential equation in D = d/dz with coefficients in the polynomial ring Q[z}.
(iv) Growth condition. There is a constant C < oo
such that gn := lcm(gi,g25 • • • ?9n) < Cn for all
n > 1.
/ KA n(~\ _ V^ n u ~n.
^To
see [Andre 1989, Theorem D, p. 14].
Definition 5.2. A special value of a G-function deo v e r K is a v a l u e
/( 6 )> w h e r e b e K' w h i c h i s
obtained by analytic continuation along some path
from
° t o b t h a t avoids s i n § u l a r Points'
fined
Siegel [1929] introduced G-functions and observed
that irrationality results could be proved for their
"special values", but did not give any details. Bombieri [1981] developed the theory of G-functions and
gave explicit irrationality criteria in specific cases
(his Theorem 6) for points close to the center of the
circle of convergence of the G-series, as a by-product
of very general results.
It is easy to show that each BBP-number is a
special value of a power series on Q that satisfies
conditions (i)-(iii) of a G-series. They do not always
satisfy the growth condition (iv), however, and in a
subsequent result we give necessary and sufficient
conditions for the condition (iv) to hold.
Lagarias: On the Normality of Arithmetical Constants
Theorem 5.3. Let R(x) = p(x)/q(x) e Q(x) with
p(x), q(x) € Q[x], where (p(x),q(x)) = 1 and q{n) ^
0 for all n > 0. /Set
and let fp(z) be the p-adic power series obtained by
interpreting p(n)/q(n) e Q C Q p . Then the power
series f(z) satisfies a homogeneous linear differential equation in d/dz with coefficients in Q[z], and
f(z) has positive radius of convergence in C and
fp(z) has a positive radius of convergence in Cp for
all primes p.
363
Theorem 5.4. Let R(x) = p(x)/q(x) e Q(x) with
p(x), q{%) € Q[x] with (p(x),q(x)) = 1 and with
g(n) ^ 0 /or a// n > 0, and set
Then the power series f(z) is a G-series {necessarily
defined overQ) if and only ifq(x) factors into linear
factors in Q[x\.
Proof
- Suppose first that q(x) factors into linear fac^ ' sa^
JL
q{x) = A[{L3{x),
s over
i=i
Proof. For the jirst assertion, let p{x) = ^ . = 0 a^
and q(x) = Y,™=0 bjx3 • T h e n t h e operator
J
~°
. ^ = ljX + m. w i t h x^m. r e l a t i v e l y p r i m e
integers. To show f{z) is a G-series, by Theorem 5.3
it suffices to we check the growth condition (iv).
where L
lcm{qi,q2,...,qn)<lcm(q(l),q(2),...,q(n))
has the property that
(5-2)
J_
<\A\l[lcm(Lj(l),...,Lj(n))
Df(z) = 0.
(5-1)
j=1
where Lj(n) — ljX + rrij. Now
Indeed one has
loglcm[l,2,...,m] = ^ l o g p
z
where a^ are defined by the polynomial identity
z
= V^ A(n) = m + o(m)
^
by the prime number theorem. This yields
z
E *i* = E a'i (I) •
lcm[l, 2,..., m] = e-^^ 1 ))
as m ^ oo. This gives a bound
4 1
Multiplying this rational function by (I-*)' " yields
a polynomial of degree / in z, which is annihilated
by dl+1/dzw, and this verifies (5-1).
For the second assertion, the power series expansion of f(z) clearly has radius of convergence 1 in C.
It is easy to establish that the the p-adic series fp(z)
has a positive radius of convergence on some p-adic
disk around zero since \q(n)\ < cnd cannot contain
more than cdlogn factors of p.
•
We now give necessary and sufficient conditions for
c
a power series arising from a BBP-number to be a
G-series.
l c m ^ l ) , . . . , L^n)) < lcm(l, 2 , . . . , \l3\n + \mj\)
nj , ,,
hn , n^
e
Substituting this in (5-2) implies condition (iv).
For the opposite direction, we will show that if
q(x) does not factor into linear factors over Q then
condition (iv) does not hold. Nagell [1922] showed
that if q(x) e Z[x] is an irreducible polynomial of
degree d > 2, then there is a positive constant c(d)
with the property that for any e > 0 there is a pos...
, , ^( x
, ,, ,
ltive constant C(e) such that
lcm(g(l), g(2),..., q(n)) > C(e)n{c{d)-£)n
(5-3)
364
Experimental Mathematics, Vol. 10 (2001), No. 3
holds for all n > 1. One can prove this result
with c(d) = (d—l)/d2. Such a lower bound applies to any denominator q(x) that does not split
into linear factors over Q. To complete the argument one must bound the possible cancellation between the numerators p(n), and denominators q(n).
If (p(x)*q(x)) = 1 over Zfxl, then
\r\ y?iv //
L JJ
Theorem 5.5. Let R(x) = p(x)/q(x) G Q(x) with
p(x), q(x) G Q[x], where (p(x),q(x)) = 1 andq(n) ^
0 for all n > 0. Set
oo
f(z) = V^
zn.
n
n=o ^ '
,
x
,
.
,
,.
,.
, ,.
, ,
^
T£
£
y^ g ^ j factors into distinct linear factors over (y),
£/ien /or each rational r in the open disk of convern
n
TT gcd(p(j),q(j)) < C ,
(5-4) gence of q(z) around z = 0 the special value f(r) is
3=1
either rational or transcendental. Furthermore there
is an effective algorithm to decide whether f(r) is
for a finite constant C = C (p(x), q(x)). This follows
mUonal
Qr transcendental.
since
A( ( \ ( W <? r<
Proof. We only sketch the details, since a similar
gcd{p(n), q{n)) < C
^ ^ h & g b e e n o b t a i n e d i n [ A d h i k a r i e ta l - > 2001].
See also
holds for all n, for a suitable C. To see this, factor
[Tijdeman > 2001, Theorem 6].
p(x) = n(z-«i) and q(x) = YKx-Pj), with a, ± fr BV expanding R(x) in partial fractions, under the
for all i,j. Then, over the number field K spanned
hypothesis that q(x) splits in linear factors over Q
one
by these roots
obtains an expansion of the form
s
ideal-gcd((n-ai), (n-0j))
| (<*-&).
R(x) = po(x) + V - ^ — ,
3= 1
Taking a norm from K/Q of the product of all these
ideals gives the desired constant C.
•
Remarks. 1. It is an interesting open question to
determine what is the largest value of c(d) allowed
in (5-3). One can show that it cannot be larger than
d-1.
2. There are many more G-functions defined over
Q than those given in Theorem 5.4. The set of Gfunctions defined over Q is closed under multiplier
tion, so that (log(l - z)f is a G-function, but its
power series coefficients around z = 0 are not given
by a rational function. Also, for rational a, b, c the
Gaussian hypergeometric function
00
f \ (IA
F
(a,
fo,
c,
z)
=
^2
/r |^n?
2 1
yc)nTl'
n=o
n c j-- which
-L- x,is-not±.
±\. above
i. kind
i • Jfor
£
is a G-function
of cthe
.„ ,
u
rA i / .,nn ^i
genenc a,b,c\ see [Andre 1996 .
According to the results of Section 4, the conclusion
of Hypothesis A is really a statement about irrational BBP-numbers. A good deal is known about
the irrationality or transcendence of the special values of the G-series covered in Theorem 5.4, a topic
that we now address.
X
^
inwhich
P^x) e ^ N > and each Cj,rj e Q. In
fact Tj $. Z> 0 , so all denominators q(n) are nonzero.
Now if r,- =Pj/qj then one has a decomposition
v ^ 1 3• _ , \ . v ^ R lrk_ (i _ „ _VI_ 27rifc\
^ n " r / - ^ ^ + Z . ^ l o g ^ zexp ^ j ,
i n w h i c h p.^ i s a p o l y n o m i a i w i t h r a t ional coeffid e n t s a n d t h e p.fca r e e f f e c t i v e i y computable algeb r a i c n u m b e r s in the field Q(exp(27ri/g,)). It follows
^
f
from
w i t h vat[onal
o ft h e
n
thig t h a t
one can express t h e
function
a g a finite s u m o f t e r m s o f t h e form a . / ( 1
form
_ zy
coefficients plus a finite sum of terms
_&.fcl o g ( 1 _ ^
the ^
a n d Q. be.
i g effectively computable algebraic numbers. The
nonlogarithmic terms all combine to give a rational
function Ro(z) with coefficients in Q. Given a rational r with 0 < r\
• < '1, it follows that /,( r ), is
a finite sum of linear forms in logarithms with al,
, ,
gebraic coefficients, evaluated at algebraic points.
Using Baker's transcendence result on linear forms
in logarithms [1975, Theorem 2.1], f(r) is transcendental if and only if the sum of all the logarithmic
terms above is nonzero. There is also an effective
decision procedure to tell whether this sum is zero
or not. If the logarithmic terms do sum to zero, then
Lagarias: On the Normality of Arithmetical Constants
365
the remaining rational function terms sum up to the
rational number /(r) = Ro(r).
•
are Lebesgue measure and measures supported on finite sets that are periodic orbits of both Ta and Tb.
The case where q(x) factors into linear factors over
Q but has repeated factors is not covered in the
result above. This case includes the polylogarithm
Lik(z) = J2^Lizn/nk
°f o r der fc, for each k > 2.
Various results are known concerning the irrationality of such numbers. For example, Li fc (l/6) is irrational for all sufficiently large integers b; see [Bombieri 1981]. In fact it is known that the set of numbers
1, Lii(p/g), . . . , Lin(p/q), with Lii(z) = log(l—z),
are linearly independent over the rationals whenever
|p| > 1 and \q\ > (4n) n ( n ~ 1 ) |p| n , according to [Nikishin 1979]. For polylogarithms one has Li /c (l) =
C(fc), also on the boundary of the disk of convergence. It is not known whether ((k) is irrational
for odd k > 5, although a very recent result of T.
Rivoal [2000] shows that an infinite number of ((k)
for odd k must be irrational.
Various results concerning this conjecture appear
in [Rudolph 1990; Parry 1996; Host 1995; Johnson 1992]. In particular, if there is any exceptional
invariant measure violating the conjecture, it must
have entropy zero with respect to Lebesgue measure,
Furstenberg's conjecture involves some ingredients
similar to Hypothesis A, and its conclusion involves
a dichotomy similar to that in Hypothesis A. This
makes it natural to ask if there is any relation between the two conjectures. At present none is known,
in either direction.
One may look for BBP-numbers 0 0 Q which
have properties similar to that expressed in the hypothesis of Furstenberg's conjecture, i.e., which possess BBP-expansions to two multiplicatively independent bases. It is known that there exist irrational BBP-numbers 0 = Yl™=i R(n)b~n which do
possess BBP-expressions to two multiplicatively independent bases. For example, Bailey and Crandall
observe that 0 = log 2 has this property, on taking
6. INVARIANT MEASURES AND FURSTENBERG'S
CONJECTURE
It is well known that for single expanding dynamib — 2 and R(x) = —,
cal system, such as the 6-transformation T& , there
always exist chaotic orbits exhibiting a wide range
6
of pathology. For example, there exist uncountably
b = 6 and H[x) — 2X — \ '
many 9 € [0,1] whose 2-transformation iterates {*„} g e e [ B & i l e y & n d Cr&ndall 2 0 0 1 > e q s ( 4 ) ? ( 1 0 ) ] T h e y
satisfy T5 < xn < ^ for all n > 0; see [Pollington
a l g o o b g e r v e t h & t Q = ^2 h a g t h i g p r o p e r t y ) a s i t
1979]. One can obtain ergodic invariant measures of p o s g e s s e s B BP-expansions to bases b = 2 and b = 3 4 ,
Tb supported on the closure of suitable orbits, which
t h e ^ ^ Q n e found b y B r o a d h u r s t [ 1 9 9 9 > e q ( 2 1 2 ) >
for example may form Cantor sets of measure zero.
ori
If one considers instead two ^-transformations, say
Tbl and Tb2, with multiplicatively independent val- Question. Do all BBP-numbers which are special values (this means they generate a nonlacunary com- u e s o f ^-functions have BBP-expansions in two mulmutative semigroup S - <T 6l ,T 62 », then the set
tiplicatively independent bases?
of ergodic invariant measures for the whole semiTo make tighter a possible connection between the
group is apparently of an extremely restricted form.
two conjectures, one can ask for which numbers does
Furstenberg has proposed the following conjecture,
the following weaker version of Hypothesis A hold.
suggested as an outgrowth of his work on topolog. . .
^
„^^
. , ,
. ,
,. T , n T. .
Invariant Measure Hypothesis. Every BBP-number to
rTn
i n _ c
ical dynamics Jburstenberg 1967, bection IV . It is
,
, , 7 ,
'
.
.
.
r
, , ,.
. ,
base b has b-transformation
iterates \x
that are
r .,,
v OAnn ^
n\
explicitly
stated m rnv/r
Marguhs
2000, Conjecture Al4 .
. „ .. J ..
.
..
,. . .
asymptotically distributed according to some limiting
Furstenberg's Conjecture. Let a, b > 2 be multiplicameasure on [0,1].
tively independent integers. The only Borel mea- T,
,,, . ,
,. , n ,
i
,i
T1
.
It would be interesting to find extra hypotheses on a
r/
7
sures on 0,1 that are simultaneously invariant er,
.,,
,. .
, ,
,
,.,
£
1
' J
^
class of arithmetical constants under which a precise
qodic measures for
,.
, , ,. , , -, ,
,, . *
n
TT
J
^
connection can be established between Hypothesis A
Ta{x) — ax (mod 1) and Th{x) = bx (mod 1)
and Furstenberg's conjecture.
366
Experimental Mathematics, Vol. 10 (2001), No. 3
7. CONCLUDING REMARKS
-,,
,
- .,,
,. -,
, ,
A/r
Many of the examples or arithmetical constants arise
.,
,
r ri £ X.- J ^ J
M
as special values of G-functions denned over the ra,
, T , ,,
. , ,
„ rr
,
tionals, or at least special values of functions satisfying linear differential equations with polynomial
coefficients in Q[x]. Based on the known results,
one may empirically group these constants into three
classes, of apparently increasing order of difficulty of
establishing irrationality or transcendence results:
1. special values of G-functions f(p/q) defined over
the rationals, with p/q inside the disk of convergence of the G-series;
2. "singular values" / ( I ) of such a G-function, which
are values taken at a singular point of the associated (minimal order) linear differential equation,
on the boundary of the disk of convergence of
a G-series, at which the G-expansion converges
absolutely; and
3. "renormalized singular values",which are the constant terms in an asymptotic expansion of a Gfunction around a singular point.
In this hierarchy, an arithmetical constant may occur as more than one type. For example, TT2/6 =
C(2) = Li 2 (l) occurs as a number of type 2, but it
is also realized as a number of type 1, which falls in
the class of constants considered in this paper. It is
a nontrivial problem to determine what is the lowest
level in the hierarchy a given constant belongs.
Various constants of types 1 and 2 appear in the
renormalization of massive Feynman diagrams, see
[Broadhurst 1999; Groote et al. 1999], where Li 4 (|)
is cited as such a constant. Multiple zeta values
and polylogarithms give many examples of type 2,
see [Borwein et al. 1997; 20011. Many of the most
L
J
J
'
interesting arithmetical constants naturally arise as
°
i i !
constants of type 2 and 3. For examples, the values
/
r
Cik) = Lu(l) appear as constants of type 2, while
v J
i} : ,
\.
LL
Eulers constant appears as a type 3 renormalized
, , _.
,.
. .
.
OT
value at z — 1 of Lii(^). The problem of showing the
linear independence of all odd zeta values £(2n+l)
over the rationals has recently been of great interest
from connections with various conjectures in arithmetical algebraic geometry, see [Goncharov 2000].
Many other examples of constants of type 2 and 3
appear in [Lehmer 1975; Flajolet and Salvy 1998].
I am not aware of any irrationality or transcendence
results proved for a constant of type 3.
^
_ ,
.
.
., ,
One can extend the hierarchy above outside the
.
class of G-functions. Bombien observes that the
°^
\
'l\z) ~ /_^ n(n2 -\-l)Z
n=1
of
BBP-type, which is not a G-series, has special
value at z = 1 given by
w-i \ _ i^> -^'(0
2
T(i)
The value z — 1 lies on the boundary of the disk of
convergence of the power series for this function, and
corresponds to type (2) above. Another example is
^
V^ \ >
]_.
2
^ ~ g-7r
n=1 ^ + 1
s e e [Flajolet and Salvy 1998, p. 18], where many
other interesting examples are given.
The relevant special values of a rational power series for the approach of Bailey and Crandall to apply
a r e z = 1 / f t for i n t e g e r fe > 2> w h e r e t h g d i s k o f c o n .
v e r g e n c e o f t h e a s s o c i a t e d p o w e r s e r i es has radius
L Q n e observeg t h a t t h e theQry of G.functions p r o .
yides irrationality results for rational values z =
a/b:
w i t h o u t r e g a r d for w h e t h e r a = l o r n o t . T h i s s u g .
gestg t h e
following
question.
Question. Given a rational value z = a/6, with 1 <
\a\ < |6|, is there an associated dynamical system
(possibly higher dimensional) for which an analogue
of Theorem 3.3 holds, relating the dynamics of one
orbit to the ^-expansion of (9, with (3 = a/bl
, ,,
, ,
.,
,
,
A,
At present there seems to be no evidence strongly
>- ^
^
r -±i J.- i
± ±
£
favoring a particular class or arithmetical constants
. A .
,
,
r
for which Hypothesis A might be expected to hold.
^
,.
.
,.
,n
, ,, ,
ra
The discussions of Sections 5 and 6 suggest that one
. ,,
.,
., - n
.
,
might consider the following classes,
1. The largest class is the set of "special values"
of power series f(z) defined over Q at z = 1/6,
arising from solutions of Df(z) = 0 for some D G
W := Q[z,d/dz], whose power-series coefficients
an —> 0 as n —> oo. This class includes all BBPnumbers.
Lagarias: On the Normality of Arithmetical Constants
367
2. One could restrict to the subclass of special values z = 1/6 of G-functions defined over the rationals. However we know of no compelling reason
to restrict to special values of G-functions.
[Bombieri and Sperber 1982] E. Bombieri and S.
Sperber, "On the p-adic analyticity of solutions of
linear
differential equations", Illinois J. Math. 26:1
(!982), 10-18.
3. The smallest class consists of a class of arithmetical constants which satisfy extra conditions anal. .
.
.
,
.
J
r _
J
ogous to the hypotheses of Furstenberg s conjeeture. These consist of those constants which are
BBP-numbers to at least two multiplicatively independent bases. One might add the further re. .
,
.
.
.
. 1 1
o^
stnction that they also be special values of Gfunctions. As noted in Section 6, this class includes 7T2 and log 2
[Borwein et al. 1997] J. M. Borwein, D. M. Bradley, and
D J
* ' Broadhurst, "Evaluations of fc-fold Euler/Zagier
sums: a compendium of results for arbitrary k"
Electron. J. Combin. 4:2 (1997), RP5.
ACKNOWLEDGEMENTS
The author thanks E. Bombieri for helpful information concerning G-functions, and for suggesting
the argument establishing (5-4) in Theorem 5.4.
He thanks D. H. Bailey for references, and J. A.
Reeds and the referee for helpful comments. Work
on this paper was done in part during a visit to the
Mathematical Sciences Research Institute, Berkeley,
September 2000.
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Jeffrey C. Lagarias, AT&T Labs - Research, Florham Park, NJ 07932-0971, United States (jcl@research.att.com)
Received September 25, 2000; accepted in revised form February 17, 2001